Achromatism and Achromatic Lenses

Achromatic Lenses

Using a suitable combination of two or more lenses of different materials, the chromatic aberration can be minimized. A combination of two lenses is usually called an achromatic combination, an achromatic or an achromatic doublet. It is possible to correct the chromatic for only two colour. The achromatism would be ideal if the image of different colours are formed in the same position and are of same size, thus the longitudinal and lateral chromatic aberration being totally eliminated. With a convex lens , the violet rays of light come to focus at a point nearer the lens and with a concave lens and a converging incident beam of light the red rays of light meet the axis at a point nearer the lens. Hence, to have an achromatic combination of two lenses , one of th lenses should be convex made form crown glass and the other concave made from flint glass. The convex lens is of smaller focal length and the concave lens of larger focal length.

Achromatism: Removal of Chromatic aberration of Two Lenses

To remove the chromatic defect, a convex lens and a concave lens of different material with suitable focal length are combined as in the diagram below. An achromat is made by placing in contact, lenses of different materials and of suitable focal length, such that the focal length of the combination is the same for both the extreme colours. The focal length of the achromat is independent of the refractive index. Let, the lenses of the doublet have mean focal length f and f’. let us suppose that \(\mu\) and \(\mu’\) represent the mean refractive index for the material of the lens for crown glass and flint glass.

Figure: condition of achromatism

For convex lens;

f = focal length of convex lens

\(f_r\) = focal length of red colour

\(f_v\) = focal length of violet colour

\(\delta\) = deviation produced for mean colour

\(\delta_v\) = deviation produced for violet colour

\(delta_r\) = deviation produced for red colour

\(\mu\) = refractive index of the lens

\(\mu_r\) = refractive index of the red colour

\(\mu_v\) = refractive index of violet colour

\(\omega\) = dispersive power of convex lens

And for concave lens all the terminology used above for convex lens are denoted by the same symbol in prime system.

Now, Lens Maker’s formula for convex lens is

$$\frac{1}{f}=(\mu-1)\biggl(\frac{1}{R_1}+\frac{1}{R_2}\biggr)$$

$$or,\;\;\;\biggl(\frac{1}{R_1}+\frac{1}{R_2}\biggr)=\frac{1}{f(\mu-1)}\dotsm(1)$$

For red colour,

$$\frac{1}{f_r}=(\mu_r-1)\biggl(\frac{1}{R_1}+\frac{1}{R_2}\biggr)$$

Using (1) ,$$\frac{1}{f_r}=\frac{(\mu_v-1)}{f(\mu-1)}\dotsm(2)$$

Similarly for violet colour,

$$\frac{1}{f_v}=\frac{(\mu_v-1)}{f(\mu-1)}\dotsm(3)$$

And Lens Maker’s formula for concave lens is $$\frac{1}{f’}=(\mu’-1)\biggl(\frac{1}{R_1}-\frac{1}{R_2}\biggr)$$

$$or,\;\;\biggl(\frac{1}{R_1}-\frac{1}{R_2}\biggr)=\frac{1}{f’(\mu’-1)}\dotsm(4)$$

For rd colour, $$\frac{1}{f’_r}=(\mu’_r-1)\biggl(\frac{1}{R_1}+\frac{1}{R_2}\biggr)$$

$$\frac{1}{f’_r}=\frac{(\mu’_r-1)}{f’(\mu’-1)}\dotsm(5)$$

And for violet colour,

$$\frac{1}{f’_v}=\frac{(\mu’_v-1)}{f’(\mu’-1)}\dotsm(6)$$

Now, combined focal length for red colour is $$\frac{1}{F_r}=\frac{1}{f_r}+\frac{1}{f’_r}$$

$$\frac{1}{F_r}=\frac{(\mu_r-1)}{f(\mu-1)}+\frac{(\mu’_r-1)}{f’(\mu’-1)}\dotsm(7)$$

Similarly combine focal length for violet colour is

$$\frac{1}{F_v}=\frac{(\mu_v-1)}{f(\mu-1)}+\frac{(\mu’_v-1)}{f’(\mu’-1)}\dotsm(8)$$

To reduce the chromatic aberration; \(F_r = F_v\)

$$or,\;\;\frac{1}{F_r}=\frac{1}{F_v}$$

i.e. $$\frac{(\mu_r-1)}{f(\mu-1)}+\frac{(\mu’_r-1)}{f’(\mu’-1)}=\frac{(\mu_v-1)}{f(\mu-1)}+\frac{(\mu’_v-1)}{f’(\mu’-1)}$$

$$or,\;\;\frac{(\mu_r-1)-(\mu_v-1)}{f(\mu-1)}=\frac{(\mu’_v-1)-(\mu’_r-1)}{f’(\mu’-1)}$$

$$or,\;\;\frac{\mu_r-\mu_v}{f(\mu-1)}=\frac{\mu’_v-\mu’_r}{f’(\mu’-1)}$$

$$or,\;\;\-frac{(\mu_v-\mu_r)}{f(\mu-1)}=\frac{\mu’_v-\mu’_r}{f’(\mu’-1)}$$

$$or,\;\;\frac{-\omega’}{f’}=\frac{\omega’}{f’}$$ where $$\omega=\biggl(\frac{\mu_v-\mu_r}{\mu-1}\biggr)\;\;and\;\; \omega’=\biggl(\frac{\mu’_v-\mu’_r}{\mu’-1}\biggr)$$

$$\therefore\;\;\frac{\omega}{f}+\frac{\omega’}{f’}=0\dotsm(9)$$

The equation (9) gives the required condition for lenses to be free from chromatic aberration. This condition is called Achromatic condition of the lenses.

The equation (9) is valid only when f and f’ have opposite signs. Therefore, this kind of defect can’t be removed by using same kind of lenses.

References:

Adhikari, P.B, Daya Nidhi Chhatkuli and Iswar Prasad Koirala.A Textbook of Physics. Vol. II. Kathmandu: Sukunda Pustak Bhawan, 2012.

Jenkins, F.A and H.E White.Fundamental of optics. New York (USA): McGraw-Hill Book Co, 1976.

wood, R.W.Physical Optics. New York (USA): Dover Publication , 1934.